= Solution
Write $\Delta T=T_0-T_B$, $\Delta C=C_0-C_B$ and assume these scales are nonzero. The two reference <liquidus> relations imply $\Delta T=-m\Delta C$. For the <similarity variable> $\eta=z/(2\sqrt{\kappa t})$ and front $h=2\lambda\sqrt{\kappa t}$,
$$
\eta_t=-\frac{\eta}{2t},\quad\eta_z=\frac1{2\sqrt{\kappa t}},\quad h'=\lambda\sqrt{\kappa/t}.
$$
Thus a <temperature> profile $T=T_*+\Delta T\theta(\eta)$ has $T_t=-\Delta T\eta\theta'/(2t)$ and $\kappa T_{zz}=\Delta T\theta''/(4t)$. Each <heat equation> becomes $\theta''+2\eta\theta'=0$. The solute <diffusion equation> becomes $\mathcal C^{l\prime\prime}+2Le\eta\mathcal C^{l\prime}=0$, with
$$
\boxed{Le=\kappa/D.}
$$
This <Lewis number> compares thermal and solutal <diffusivities>. Inside the <mushy layer>, $T^s-T_B=-m(C^s-C_B)$ gives $\boxed{\mathcal C^s=\theta^s}$. The wall and far-field conditions become $\theta^s(0)=0$ and $\theta^l,\mathcal C^l\to0$ as $\eta\to\infty$.
At the moving <phase boundary>, equality of <temperatures> gives $\theta^s=\theta^l+(T_\infty-T_B)/\Delta T$. Equality of <concentrations> gives $\mathcal C^s=\mathcal C^l+1$. Therefore the dimensionless parameters with the convention used here are
$$
\boxed{\mathcal R_\theta=\frac{T_0-T_\infty}{T_0-T_B},\qquad \mathcal R_C=\frac{C_0}{C_0-C_B},\qquad S=\frac{\rho L\kappa}{k(T_0-T_B)}.}
$$
Neither <concentration> nor <temperature> offsets should be dropped when enforcing continuity. In particular $\mathcal R_\theta$ can be negative. If $k=\rho c_p\kappa$, then $S=L/[c_p\Delta T]$: this is the <inverse Stefan number>, although the paper calls $S$ its Stefan number.
To check every factor of two in the flux conditions, divide the thermal <Stefan condition> by $k\Delta T/(2\sqrt{\kappa t})$, and the solute balance by $D\Delta C/(2\sqrt{\kappa t})$. This gives
$$
\boxed{2\phi\lambda S=\theta^{s\prime}-\theta^{l\prime},\qquad 2\phi Le\lambda(\mathcal C^l+\mathcal R_C)=-\mathcal C^{l\prime}.}
$$
Finally $T_z^l=-mC_z^l$ and $-m\Delta C=\Delta T$ imply $\boxed{\theta^{l\prime}=\mathcal C^{l\prime}}$. This is <marginal equilibrium at a mush-liquid boundary>. Together these calculations produce all the rescaled differential equations and all three interface balances.
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